A008306 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + x + n + n x\right)\,P_{n-2}(x) + \left(x + n x\right)\,P_{n-2}'(x) + \left(1 + n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	9	3							12
3	44	20							64
4	265	175	15						455
5	1854	1554	210						3618
6	14833	15183	3010	105					33131
7	133496	161160	41552	2520					338728
8	1334961	1857231	591885	53550	945				3838572
9	14684570	23130470	8777010	1051820	34650				47678520
10	176214841	309978823	136508856	20436955	1022175	10395			644172045
11	2290792932	4451091996	2231644272	401054940	26966940	540540			9402091620
12	32071101049	68214438279	38364945619	8051051008	682692010	21126105	135135		147405489205
13	481066515734	1111703812842	693153586034	166416810560	17060143100	718707990	9459450		2470129035710
14	7697064251745	19203056856735	13146942917550	3554789403165	427869812370	22922349450	472972500	2027025	44053120590540

Technical details

Node id: n-2b81ed5ecd810764 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-ac32f0aa28a8fb19

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        2;
  2:        9,        3;
  3:       44,       20;
  4:      265,      175,      15;
  5:     1854,     1554,     210;
  6:    14833,    15183,    3010,     105;
  7:   133496,   161160,   41552,    2520;
  8:  1334961,  1857231,  591885,   53550,   945;
  9: 14684570, 23130470, 8777010, 1051820, 34650;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(n+1)*x)*P(n-2,x) + (n+1)*x*d/dx P(n-2,x) + (n+1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (n + 1)*x)*P[n - 2, x] + (n + 1)*x*D[P[n - 2, x], x] + (n + 1)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (n + 1)*x)*v[i - 2] + (n + 1)*x*deriv(v[i - 2], x) + (n + 1)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows; related to number of preorders.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741